Alexandra Fuller, known universally as Bobo, grew up on a succession of ever-poorer tobacco and cattle farms in Rhodesia, Malawi and Zambia in the 1970s and 80s, with alcoholic parents, a manic-depressive mother, civil war going on all around, and three out of her four beloved siblings dying in infancy. She noticed quite early on that she was the wrong colour for Africa: a "marshmallow".

Now she has published a childhood memoir, Don't Let's Go to the Dogs Tonight, that is being compared to the Diary of Anne Frank; to the works of Michael Ondaatje and Arundhati Roy. Richard E Grant has described it as "hilarious and insane". You can hear it this week on BBC Radio 4's Book of the Week. Several friends have told me about how they wept to hear of Fuller's sister, Olivia, drowning as a toddler; Fuller, a couple of years older, had been told, "Keep and eye on her," and has never quite escaped blaming herself.

Swooping into a swanky London club, Fuller doesn't in the least resemble the "worm-bellied, mud-spattered sprog tearing round the farm on a motorbike" that she used to be. She is sunkissed, beautiful, hungover ("I hope none of the people in here were in last night," she says, peering around in trepidation) and astounded by the reception her book has had - translated into 12 languages, a six-week world tour, and the rest.

All good? No accusations of being an apologist for racist imperialism? "Well, at a reading in the States I had some of the: 'What right have you to write about Africa, privileged white girl?' You know. I just said no one has a right to write, it isn't a choice. I have eight or nine novels musting on a shelf and in the end I had to write the truth as I saw it."

Fuller was conceived beside Victoria Falls, born in Derbyshire, and at the age of two moved back to Africa. Both her parents had lived in Africa for years. "My father came out to Kenya as a young man to see a giraffe and escape alcoholic relatives. Mum was living the life of a colonial beauty in the twilight of Kenya's empire days, and my grandfather had built a church there, which burned down. We weren't imperialist bureaucrats, out for a year or two and then going back to Britain and saying, 'Oh, yes, when I was in Zambia...' We were settlers."

She isn't defensive about this at all - you get the feeling that she has been thinking about it all her life: how it is to belong to a place that does not belong to you. Her parents are still in Africa, in a "rudimentary dwelling on the banks of the Zambezi". Her surviving sibling, Vanessa, "lives in a rock house balanced on top of a kopje [rocky outcrop] near the Kafue river [in Zambia]". Fuller has been living in Wyoming (loves the landscape, not sure about the mindset) since 1994 with her husband and two children, but is moving back to Africa this year - to Tanzania.

The Rhodesian (as was) civil war was the background to her life. She grew up with conflict, hunger and disease constantly at the door (or doors, rather, as their farms were repossessed and they ended up in an area marked on the old maps "NotFitForWhiteMan'sHabitation"). Her father would set off into the bush for days on end to fight the "terrorist guys"; her mother was meanwhile rounding up wild cattle, abusing squatters, and shooting a cobra in the pantry.

The baby before Bobo, Adrian, died ("The story changes depending on what mum is drinking"), and Fuller was born "to replace him". Then Olivia drowned; then a baby brother never came back from hospital, and her mother had a nervous breakdown. A servant practically murdered another: her father tracked him down through the bush, and the last Fuller saw of him was as he was dragged off behind a pick-up. There are minefields and malaria, and an old woman by the main road who beats a gong when a convoy goes past, so the rebels know to hide, or to attack. At seven she was stripping down machine guns, her dad telling her: "You'd have to do it faster than that."

Not surprisingly, "I never felt immortal - always a breath away from dying, and that gives it a supernatural clarity. There was so little stability you had no time to heal, no way to forget. And longing makes you remember." She recognises that: "You wouldn't wish alcoholic parents, a manic-depressive mother and war on anyone. But to know the fabulous resilience of human beings, and the value of being alive?"

Here is Fuller's mother's conversational repertoire with a new guest, as reported in the book: (1) the war; (2) dead children; (3) insanity; and (4) being Nicola Fuller of central Africa. Here is the guest's repertoire of reaction: (1) delight; (2) mild intoxication coupled with growing disbelief; (3) extreme intoxication coupled with growing panic; and (4) loss of consciousness. Young Bobo falls asleep to the sound of her mother saying, "We were prepared to die, you see, to keep one country white-run," and wakes to find her staring into the dawn with a warm beer still in her hands, and the guest passed out on the lawn. "We're all mad," says her mother some chapters later, "but only I have the certificate to prove it."

Did her mother like the book? "She said: 'Elspeth Huxley wrote a very nice book about her family.' She felt betrayed. I told her more people would have been more betrayed if I hadn't written the truth. I love my mother so much, because I see the whole of her. People think the book is a love-letter to Africa but really it's a love-letter to her."

And how do they reconcile, these ferociously racist parents and a daughter who, when she heard that Nelson Mandela had been released, rang her boyfriend and told him: "Go and buy all the beer in the shop, Mandela is free!"?

"I was so so happy: I was in Canada, and the washing was frozen on the line. I thought: 'It can all be over, everything can be all right, I won't have to be a marshmallow any more.' I'll never forget the moment that I heard." And the parents? "I don't judge them. They live more purely and in touch with Africa than patronising liberals. They say stupid things but they live so close to the earth, so close to the wind, so African, that it's pointless. My mother said recently: 'All of us are guests, not just in Africa but on the planet.' So..."

Her love for her mother is beyond worry, blame or forgiveness. "You never stop longing for your mother if she's inaccessible to you - as mine was, for long periods, through drink and madness. I used to sniff her handkerchief at boarding school - the smell of Vicks and tea." (Boarding school was a mixed relief from home life. After Mugabe was elected in 1980, the other white children all left, almost overnight, and for the first time Fuller met a black person with a surname: a fellow pupil, Oliver Tendai Chiswe.)

"I always knew mum loved me - tough, look-after-yourself love, as if she knew she wouldn't always be there." This was a woman who would make her daughter ride all day without water, then cut off her own hair to wrap it round bits of meat to tempt an injured owl to eat. Fuller remembers all these details: even the sound of her mother rubbing her nyloned feet against each other during a dull shift as a police reservist, for example. Does she really remember that sound? She smiles with recognition as the memory pops back again: "Yes! Because if she was doing that, she was happy."

I suggest that there are those who may think it a little premature to write a memoir at 31. Fuller cuts in: "Do you know what life expectancy is in Zimbabwe? Thirty-seven. I'm an old lady. It took me 30 years and a brilliant education to find my voice. And I wrote this in Wyoming - I couldn't write it at home. There it's as much as anyone can do to survive the day."

So what about Mugabe, and the election? "Do you know that his son died at three? His name in Shona meant 'Suffering of the Country'. Mugabe wasn't allowed out of prison for the funeral. What does that do to you? He was a brilliant man. A brilliant man prostituted to power, and now he's going to his grave and wants to take the country with him. It's a corruption which grows out of a sense of entitlement."

Don't Let's Go to the Dogs Tonight is a child's-eye view, a "babbling", Fuller calls it, but she has her adult view of the country she loves, and we may see it in print yet. "Thirty-seven," she says again, and her sadness is palpable.

· Don't Let's Go to the Dogs Tonight is published by Picador, price £15.99.